IEEE
Proof
Irregular Trellis for the Near-Capacity Unary
Error Correction Coding of Symbol
Values From an Infinite Set
12
3
Wenbo Zhang, Student Member, IEEE, Matthew F. Brejza, Tao Wang, Robert G. Maunder, Senior Member, IEEE,
and Lajos Hanzo, Fellow, IEEE
4 5
Abstract—Irregular joint source and channel coding 1
(JSCC) scheme is proposed, which we refer to as the irregu-2
lar unary error correction (IrUEC) code. This code operates on 3
the basis of a single irregular trellis, instead of employing a set 4
of separate regular trellises, as in previous irregular trellis-based 5
codes. Our irregular trellis is designed with consideration of the 6
UEC free distance, which we characterize for the first time in 7
this paper. We conceive the serial concatenation of the proposed 8
IrUEC code with an irregular unity rate code (IrURC) code and 9
propose a new EXtrinsic Information Transfer (EXIT) chart 10
matching algorithm for parametrizing these codes. This facil-11
itates the creation of a narrow EXIT tunnel at a low Eb/N0 12
value and provides near-capacity operation. Owing to this, 13
our scheme is found to offer a low symbol error ratio (SER), 14
which is within 0.4 dB of the discrete-input continuous-output 15
memoryless channel (DCMC) capacity bound in a particular 16
practical scenario, where gray-mapped quaternary phase shift 17
keying (QPSK) modulation is employed for transmission over 18
an uncorrelated narrowband Rayleigh-fading channel with an 19
effective throughput of 0.508bit s−1Hz−1. Furthermore, the 20
proposed IrUEC–IrURC scheme offers a SER performance gain 21
of 0.8 dB, compared to the best of several regular and irregular 22
separate source and channel coding (SSCC) benchmarkers, 23
which is achieved without any increase in transmission energy, 24
bandwidth, transmit duration, or decoding complexity. 25
Index Terms—Joint source–channel coding, irregular codecs, 26
channel capacity, iterative decoding. 27
I. INTRODUCTION 28
I
N MOBILE wireless scenarios, multimedia transmission is 29required to be bandwidth efficient and resilient to transmis-30
sion errors, motivating both source and channel coding [1]–[3]. 31
Classic Separate Source and Channel Coding (SSCC) may 32
be achieved by combining a near-entropy source code with a 33
near-capacity channel code. In this scenario, it is theoretically 34
Manuscript received October 31, 2014; revised April 14, 2015 and July 25, 2015; accepted October 8, 2015. This work was supported in part by the EPSRC, Swindon UK under Grant EP/J015520/1 and Grant EP/L010550/1, in part by the TSB Swindon UK under Grant TS/L009390/1, in part by the RCUK under the India-UK Advanced Technology Centre (IU-ARC), and in part by the EU under the CONCERTO project and in part by the European Research Council’s Advanced Fellow grant. The associate editor coordinating the review of this paper and approving it for publication was M. Xiao.
The authors are with School of Electronics and Computer Science, University of Southampton, Southampton SO17 1BJ, U.K. (e-mail: [email protected]. ac.uk; [email protected]; [email protected]; [email protected]. ac.uk; [email protected]).
Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org.
Digital Object Identifier 10.1109/TCOMM.2015.2493149
possible to reconstruct the source information with an infinites- 35 imally low probability of error, provided that the transmission 36 rate does not exceed the channel’s capacity [4]. However, sep- 37 arate source-channel coding [4] is only capable of approaching 38 the capacity in the general case by imposing both infinite com- 39 plexity and infinite latency. For example, adaptive arithmetic 40 coding [5] and Lempel-Ziv coding [6] are capable of encoding 41 a sequence of symbols using a near-entropy number of bits per 42 symbol. However, these schemes require both the transmitter 43 and receiver to accurately estimate the occurrence probability 44 of every symbol value that the source produces. In practice, the 45 occurrence probability of rare symbol values can only be accu- 46 rately estimated, if a sufficiently large number of symbols has 47 been observed, hence potentially imposing an excessive latency. 48 This motivates the design of universal codes, such as the 49 Elias Gamma (EG) code [7], which facilitate the binary encod- 50 ing of symbols selected from infinite sets, without requiring 51 any knowledge of the corresponding occurrence probabilities 52 at either the transmitter or receiver. The H.264 video codec 53 [8] employs the EG code and this may be concatenated with 54 classic channel codes, such as a Convolutional Code (CC) to 55 provide a separate error correction capability. Nevertheless, this 56 SSCC typically suffers from a capacity loss, owing to the resid- 57 ual redundancy that is typically retained during EG encoding, 58 which results in an average number of EG-encoded bits per 59 symbol that exceeds the entropy of the symbols. 60 In order to exploit the residual redundancy and hence to 61 achieve near-capacity operation, the classic SSCC schemes 62 may be replaced by Joint Source and Channel Coding 63 (JSCC) arrangements [9] in many applications. As we have 64 previously demonstrated in [10, Fig. 1], the symbols that 65 are EG encoded in H.264 are approximately zeta probabil- 66 ity distributed [11], resulting in most symbols having low 67 values, but some rare symbols having values around 1000. 68 Until recently, the decoding complexity of all previous JSCCs, 69 such as Reversible Variable Length Codes (RVLCs) [12] and 70 Variable Length Error Correction (VLEC) codes [13], increased 71 rapidly with the cardinality of the symbol set, so much so that it 72 became excessive for the H.264 symbol probability distribution 73 and asymptotically tending to infinity, when the cardinality is 74
infinite. 75
Against this background, a novel JSCC scheme referred to 76 as a Unary Error Correction (UEC) code [10] was proposed 77 as the first JSCC that mitigates the capacity loss and incurs 78 0090-6778 © 2015 IEEE. Personal use is permitted, but republication/redistribution requires IEEE permission.
IEEE
Proof
Fig. 1. Schematic of the proposed IrUEC-IrURC scheme, in which an IrUEC code is serially concatenated with IrURC code and Gray-coded QPSK modulation schemes. Here,π1andπ2represent interleavers, whileπ1−1andπ2−1represent the corresponding deinterleavers.
only a moderate decoding complexity, even when the cardi-79
nality of the symbol set is infinite. In a particular practical 80
scenario, an iteratively-decoded serial concatenation of the 81
UEC code with an Irregular Unity Rate Code (IrURC) was 82
shown to offer a 1.3 dB gain compared to a SSCC bench-83
marker, without incurring an increased transmission energy, 84
duration, bandwidth or decoding complexity. Furthermore, this 85
was achieved within 1.6 dB of the Quaternary Phase Shift 86
Keying (QPSK)-modulated uncorrelated narrow band Rayleigh 87
fading Discrete-input Continuous-output Memoryless Channel 88
(DCMC) capacity bound. 89
In this paper, we will further exploit the properties of UEC 90
codes in order to facilitate reliable operation even closer to 91
the capacity bound. More specifically, we propose an Irregular 92
Unary Error Correction (IrUEC) code, which extends the regu-93
lar UEC of our previous work [10]. This IrUEC code employs 94
different UEC parametrizations for the coding of different sub-95
sets of each message frame, in analogy with previous irregular 96
codes, such as the IrURC [14], the Irregular Convolutional 97
Code (IrCC) [15] and the Irregular Variable Length Code 98
(IrVLC) [16]. However, these previous irregular codes oper-99
ate on the basis of a number of separate trellises, each of which 100
has a different but uniform structure and is used for the cod-101
ing of a different subset of the message frame. By contrast, our 102
new IrUEC code operates on the basis of a single irregular 103
trellis having a novel design. This trellis has a non-uniform 104
structure that applies different UEC parametrizations for dif-105
ferent subsets of the frame on a bit-by-bit basis. This allows 106
the irregularity of the proposed IrUEC code to be controlled 107
on a fine-grained bit-by-bit basis, rather than on a symbol-by-108
symbol basis, hence facilitating nearer-to-capacity operation. 109
More specifically, our results demonstrate that controlling the 110
IrUEC irregularity on a bit-by-bit basis offers gains of up to 111
0.2 dB over the symbol-by-symbol approach, without impos-112
ing any increase in transmission energy, bandwidth, latency or 113
decoding complexity. 114
This bit-by-bit IrUEC approach is facilitated by some partic-115
ular properties of UEC codes, which grant some commonality 116
to all UEC parametrizations. By exploiting this fine-grained 117 control of the IrUEC irregularity, the IrUEC EXtrinsic 118 Information Transfer (EXIT) function may be shaped to cre- 119 ate a narrow, but marginally open EXIT chart tunnel. This 120 implies that near-capacity operation is facilitated, according to 121 the theoretical properties of EXIT charts [17]. 122 The rest of this paper is organized as follows. Section II 123 describes a transmitter that serially concatenates the proposed 124 IrUEC encoder with a IrURC encoder, while Section III 125 describes the corresponding iterative receiver. The IrUEC 126 encoder and decoder operate on the basis of our novel irregular 127 trellis structure, which allows bit-level control of the irregular 128 coding fractions. The free distance of UEC codes is quantified 129 for the first time in Section IV, which proposes a novel low- 130 complexity heuristic method conceived for this purpose. This is 131 used for selecting a family of UEC trellis structures having a 132 wide variety of EXIT function shapes. The resultant UEC trel- 133 lis family maximises the design freedom for the IrUEC EXIT 134 function and therefore has a general applicability for IrUEC 135 codes used in diverse applications. Furthermore, for any partic- 136 ular application of an IrUEC code, we propose a double-sided 137 EXIT chart matching algorithm for selecting the specific frac- 138 tion of the frame that should be encoded using each IrUEC 139 and IrURC trellis structure. This allows the EXIT functions of 140 IrUEC and IrURC codes to be accurately shaped for closely 141 matching each other, hence creating a narrow but marginally 142 open EXIT chart tunnel. In Section V, the proposed IrUEC- 143 IrURC scheme is compared to an irregular JSCC benchmarker, 144 which is referred to as the EG-IrCC-IrURC scheme. The first 145 version of this benchmarker employs the recursive systematic 146 CCs that were originally recommended as IrCC component 147 codes in [15]. However, we demonstrate that the systematic 148 nature of these IrCC component codes results in a capacity loss. 149 This motivates the employment of the second version of our 150 EG-IrCC-IrURC benchmarker, which employs the recursive 151 non-systematic CCs of [10] as the IrCC component codes. The 152 simulation results of Section V show that in a particular prac- 153 tical scenario, the proposed IrUEC-IrURC scheme provides a 154
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Proof
0.8 dB gain over the best SSCC benchmarker, while operating155
within 0.4 dB of the capacity bound. This is achieved with-156
out any increase in transmission energy, bandwidth, latency or 157
decoding complexity. Finally, Section VI concludes the paper. 158
II. IRUEC-IRURC ENCODER 159
In this section, we introduce the transmitter of the proposed 160
IrUEC-IrURC scheme of Fig. 1. The IrURC encoder employs 161
T number of component Unity Rate Code (URC) encoders 162
{URCt}T
t=1, each having a distinct independent trellis structure. 163
By contrast, the IrUEC employs a unary encoder and a novel 164
Irregular Trellis (IrTrellis) encoder with a single irregular trel-165
lis. However, in analogy with the IrURC code, we note that this 166
irregular trellis comprises a merging of S component UEC trel-167
lis structures{UECs}sS=1, where UECs is the s-th component 168
UEC trellis structure that is defined by the corresponding code-169
word set Cs, as illustrated in [10, Fig. 3(a)]. In Section II-A 170
and Section II-B, the two components of the IrUEC encoder 171
in Fig. 1, namely the unary encoder and the novel IrTrellis 172
encoder are detailed. The IrURC encoder and the modulator are 173
introduced in Section II-C. 174
A. Unary Encoder 175
The IrUEC encoder is designed for conveying a vector 176
x= [xi]ai=1 comprising a number of symbols, as shown in 177
Fig. 1. The value of each symbol xi ∈ N1 may be modeled 178
by an Independent and Identically Distributed (IID) Random 179
Variable (RV) Xi, which adopts the value x with a prob-180
ability of Pr(Xi = x) = P(x), where N1= {1, 2, 3, . . . , ∞} 181
is the infinite-cardinality set comprising all positive integers. 182
Throughout this paper we assume that the symbol values obey a 183
zeta probability distribution [11], since this models the symbols 184
produced by multimedia encoders, as described in Section I. 185
The zeta probability distribution is defined as 186 P(x) = x −s ζ(s), (1) whereζ(s) =x∈N 1x
−s is the Riemann zeta function, s> 1 187
parametrizes the zeta distribution and p1= Pr(Xi = 1) = 188
1/ζ(s) is the probability of occurrence for the most frequently 189
occurring symbol value, namely x = 1. Without loss of gener-190
ality, Table I exemplifies the first ten symbol probabilities P(xi) 191
for a zeta distribution having the parameter p1= 0.797, which 192
corresponds to s= 2.77 and was found in [10] to allow a fair 193
comparison between unary- and EG-based schemes. Note that 194
other p1 values of 0.694, 0.8 and 0.9 have been investigated 195
in [18], [19]. In the situation where the symbols obey the zeta 196
probability distribution of (1), the symbol entropy is given by 197 HX= x∈N1 H [P(x)] = ln(ζ(s)) ln(2) − sζ(s) ln(2)ζ(s), (2) where H [ p]= p log2(1/p) and ζ(s) = −
x∈N1ln(x)x
−s is 198
the derivative of the Riemann zeta function. 199
As shown in Fig. 1, the IrUEC encoder represents the source 200
vector x using a unary encoder. More specifically, each symbol 201
TABLE I
THEFIRSTTENSYMBOLPROBABILITIES FOR AZETADISTRIBUTION HAVING THEPARAMETERp1= 0.797,ASWELL AS THECORRESPONDINGUNARY ANDEG CODEWORDS
xi in the vector x is represented by a corresponding codeword 202 yi that comprises xi bits, namely(xi− 1) binary ones followed 203
by a zero, as exemplified in Table I. When the symbols adopt 204 the zeta distribution of (1), the average unary codeword length l 205 is only finite for s> 2 and hence for p1> 0.608 [10], in which 206
case we have 207
l=
x∈N1
P(x) · x =ζ(s − 1)
ζ(s) . (3)
Note that for p1≤ 0.608, our Elias Gamma Error Correction 208 (EGEC) code of [19] may be employed in order to achieve a 209 finite average codeword length, albeit at the cost of an increased 210 complexity. In our future work, we will consider a novel 211 Irregular EGEC code, which has a finite codeword length for 212 p1≤ 0.608. Without loss of generality, in the example scenario 213 of p1= 0.797, an average codeword length of l = 1.54 results. 214 The output of the unary encoder is generated by concatenating 215 the selected codewords{yi}ai=1, in order to form the b-bit vec- 216
tor y= [yj]bj=1. For example, the source vector x= [4, 1, 2] of 217 a= 3 symbols yields the b = 7-bit vector y = [1110010]. Note 218 that the average length of the bit vector y is given by(a · l). 219
B. IrTrellis Encoder 220
Following unary encoding, the IrTrellis encoder of Fig. 1 221 employs a single new irregular trellis to encode the bit vec- 222 tor y, rather than using a selection of separate trellis structures, 223 as is necessary for the IrCC [15], IrVLC [16] and IrURC [14] 224 coding schemes. Our novel irregular trellis structure is facil- 225 itated by the properties of the generalised trellis structure of 226 [10, Fig. 3(a)], which was the basis of our previous work on 227 regular UEC codes. This trellis structure is parametrized by 228 an even number of states r and by the UEC codeword setC, 229 which comprises r/2 binary codewords of a particular length 230 n. Each bit yj of the unary-encoded bit sequence y= [yj]bj=1 231
corresponds to a transition in the UEC trellis from the previous 232 state mj−1∈ {1, 2, . . . , r} to the next state mj ∈ {1, 2, . . . , r}. 233
Each next state mj is selected from two legitimate alternatives, 234
depending both on the previous state mj−1and on the bit value 235 yj, according to [18, (3)]. More specifically, regardless of how 236 the UEC trellis is parametrized, a unary-coded bit of yj = 1 237
causes a transition towards state mj = r − 1 or r of the gener- 238
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Proof
at the end of each unary codeword causes a transition to state240
mj = 1 or 2, depending on whether the current symbol xi has 241
an odd or even index i . 242
This common feature of all UEC trellises maintains syn-243
chronisation with the unary codewords and allows the residual 244
redundancy that remains following unary encoding to be expli-245
cated for error correction. Furthermore, this common treatment 246
of the unary-encoded bits in y between all UEC trellises allows 247
them to merge in order to form our novel irregular trellis. More 248
specifically, our novel irregular trellis can be seen as concate-249
nation of a number of individual UEC trellis structures with 250
different numbers of states r and different codeword setsC. By 251
contrast, CCs, Variable Length Codes (VLCs) and URC codes 252
having different parametrizations do not generally exhibit the 253
required similarity in their trellises. More specifically, the final 254
state of a particular component encoder has no specific relation-255
ship with the initial state of the subsequent component encoder, 256
hence preventing their amalgamation into IrCC, IrVLC and 257
IrURC trellises, respectively. 258
The IrTrellis encoder of Fig. 1 encodes the b-bit unary-259
encoded bit sequence y= [yj]bj=1 using an irregular trellis 260
that is obtained by concatenating b number of regular UEC 261
trellis structures. The proposed IrTrellis can be constructed 262
using diverse combinations of component regular UEC trel-263
lises, having any parametrization. However, the component 264
regular trellises may be strategically selected in order to care-265
fully shape the EXIT function of the IrUEC code, for the sake 266
of producing a narrow EXIT chart tunnel and for facilitating 267
near-capacity operation, as it will be detailed in Section IV. 268
Without loss of generality, Fig. 1 provides an example of the 269
irregular trellis for the example scenario where we have b= 7. 270
Each bit yj in the vector y is encoded using the correspond-271
ing one of these b trellis structures, which is parametrized 272
by an even number of states rj and the codeword set Cj = 273 {cj 1, c j 2, . . . , c j rj/2−1, c j
rj/2}, which comprises rj/2 binary code-274
words of a particular length nj. Note that successive trellis 275
structures can have different numbers of states, subject to the 276
constraint rj ≤ rj−1+ 2, as it will be demonstrated in the fol-277
lowing discussions. Note that this constraint does not restrict 278
the generality of the IrUEC trellis, since the IrUEC EXIT func-279
tion shape is independent of the ordering of the component 280
trellis structures. 281
As in the regular UEC trellis of [10], the encoding process 282 always emerges from the state m0= 1. The unary-encoded 283 bits of y are considered in order of increasing index j and 284 each bit yj causes the novel IrTrellis to traverse from the 285
previous state mj−1∈ {1, 2, . . . , rj−1} to the next state mj ∈ 286
{1, 2, . . . , rj}, which is selected from two legitimate alterna- 287
tives. More specifically, 288
mj = 1+ oddmj−1 if yj = 0 minmj−1+ 2, rj− odd mj−1 if yj = 1 , (4)
where the function odd(·) yields 1 if the operand is odd or 0 if 289 it is even. Note that the next state mj in the irregular trellis is 290
confined by the number of states rj in the corresponding trellis 291
structure, rather than by a constant number of states r , as in the 292 regular UEC trellis of [10]. In this way, the bit sequence y iden- 293 tifies a path through the single irregular trellis, which may be 294 represented by a vector m= [mj]bj=0 comprising b+ 1 state 295
values. As in the regular UEC trellis of [10], the transitions of 296 the proposed irregular trellis are synchronous with the unary 297 codewords of Table I. More specifically, just as each symbol 298 xi in the vector x corresponds to an xi-bit codeword yi in the 299
vector y, the symbol xi also corresponds to a section mi of 300
the trellis path m comprising xi transitions between (xi + 1) 301
states. Owing to this, the path m is guaranteed to terminate 302 in the state mb= 1, when the symbol vector x has an even 303
length a, while mb= 2 is guaranteed when a is odd [10]. Note 304
that the example unary-encoded bit sequence y= [1110010] 305 corresponds to the path m= [1, 3, 5, 3, 2, 1, 1, 2] through the 306
irregular UEC trellis of Fig. 2. 307
The path m may be modeled as a particular realization 308 of a vector M= [Mj]bj=0comprising (b + 1) RVs. Note that 309
the probability Pr(Mj = mj, Mj−1= mj−1) = P(mj, mj−1) 310
of the transition from the previous state mj−1to the next state 311 mj can be derived by observing the value of each symbol in 312
the vector x and simultaneously its corresponding index. The 313 state transition M= {Mj}bj=0 follows the same rule shown in 314
(4), and all the transitions can be categorised into four types, as 315 illustrated in [10, (8)]. Owing to this, the probability of a tran- 316 sition P(mj, mj−1) in the irregular trellis is associated with the 317
transition probabilities Pr(Mj = m, Mj−1= m) = P(m, m) 318
in (5), shown at the bottom of the page. Note that these 319
P(mj, mj−1) = ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ 1 2l 1− m j−1 2 x=1 P(x) if mj−1∈ {1, 2, 3, . . . , rj−1− 2}, mj = mj−1+ 2 1 2lP(x) x= m j−1 2 if mj−1∈ {1, 2, 3, . . . , rj−1− 2}, mj = 1 + odd(mj−1) 1 2l 1− r j−1 2 −1 x=1 P(x) if mj−1∈ {rj−1− 1, rj−1}, mj = 1 + odd(mj−1) 1 2l l−rj−1 2 − r j−1 2 −1 x=1 P(x) x−rj−1 2 if mj−1∈ {rj−1− 1, rj−1}, mj ∈ {rj − 1, rj} 0 otherwise (5)
IEEE
Proof
Fig. 2. An example of the proposed irregular UEC trellis, which is obtained by amalgamating seven different UEC trellises. Here, the component UEC codebooks C1= {0, 1, 1, 1}, C2= {0, 1, 1, 1}, C3= {000, 000, 000}, C4= {00, 01}, C5= {000, 011}, C6= {000, 011} and C7= {0000} are employed.
transition probabilities are generalized, allow their application 320
to any IrUEC trellis and to any source probability distribution 321
P(x). 322
Similar to the regular UEC trellis encoder, the proposed 323
IrTrellis encoder represents each bit yj in the vector y by a 324
codeword zj comprising nj bits. This is selected from the cor-325
responding set of rj/2 codewords Cj = {c1j, c2j, . . . , crj j/2−1, 326
crjj/2} or from the complementary set Cj = {c1j, c2j, . . . , 327
crj j/2−1, c
j
rj/2}, which is achieved according to 328 zj = ⎧ ⎪ ⎨ ⎪ ⎩ cmj j−1/2 if yj = odd(mj−1) cmj j−1/2 if yj = odd(mj−1) . (6)
Finally, the selected codewords are concatenated to obtain 329
the bit vector z= [zk]bk=1¯n of Fig. 1, where ¯n = 1 b
b j=1nj 330
is the average codeword length. For example, the path m= 331
[1, 3, 5, 3, 2, 1, 1, 2] through the irregular UEC trellis of Fig. 2 332
yields the encoded bit sequence z= [1000011111110000], 333
which comprises b¯n = 16 bits, where we have ¯n = 167. 334
Note that the bit vector z may be modeled as a specific real-335
ization of a vector Z= [Zk]bk=1¯n comprising b¯n binary RVs. 336
Observe in Fig. 2 that each of the b component trellis struc-337
tures in the irregular UEC trellis of the IrTrellis encoder is 338
designed to obey symmetry and to rely on complementary 339
codewords. Hence, bits of the encoded bit vector Z have 340
equiprobable values, where Pr(Zk= 0) = Pr(Zk= 1) = 0.5, 341
and the bit entropy obeys HZk = H[Pr(Zk = 0)] + H[Pr(Zk = 342
1)] = 1. Owing to this, in contrast to some of the benchmarkers 343
to be considered in Section V, the proposed IrUEC scheme of 344
Fig. 1 does not suffer from additional capacity loss. 345
We assume that each of the b trellis structures in the proposed 346 irregular UEC trellis is selected from a set of S component 347 UEC trellis structures{UECs}sS=1, corresponding to a set of S 348 component codebooks{Cs}Ss=1. More specifically, we assume 349
that each codebookCs is employed for generating a particu- 350
lar fractionαs of the bits in z, where we haveSs=1αs = 1. 351 Here, the number of bits generated using the codebookCs is 352
given by b¯n · αs. We will in Section IV show that the fractions 353 α = {αs}S
s=1 may be designed in order to appropriately shape 354
the IrUEC EXIT function. Moreover, the IrUEC coding rate is 355 given by RIrUEC=
S
s=1αs· RUECs, where the corresponding 356 coding rate RUECs of the regular UECs code depends on the 357 codebookCs and is given by [10, Eq. (11)]. 358
C. IrURC Encoder and Modulator 359
As shown in Fig. 1, the IrUEC-encoded bit sequence z is 360 interleaved in the blockπ1in order to obtain the bit vector v, 361 which is encoded by an IrURC encoder [14], [20] comprising 362 T component URC codes{URCt}Tt=1. Unlike our IrUEC code, 363 each component URC code URCt of the IrURC code employs 364 a separate trellis structure. This is necessary, since the final 365 state of each component URC code has no relation to the ini- 366 tial state of the subsequent component URC code, as described 367 in Section II-B. Therefore, the interleaved IrURC-encoded bit 368 vector u is decomposed into T sub-vectors{ut}Tt=1, each having 369
a length given by b¯n · βt, whereβt represents the specific frac- 370
tion of the bits in v that are encoded by the component URCt 371 code, which obeystT=1βt = 1. In Section IV, we also show 372 that the fractionsβ = {βt}Tt=1may be designed in order to shape 373
IEEE
Proof
In common with each of its T number of component URC375
codes, the IrURC code has a coding rate of RIrURC= 1, regard-376
less of the particular irregular code design. Owing to this, each 377
of the T number of binary sub-vectors{vt}tT=1that result from 378
IrURC encoding has the same length as the corresponding sub-379
vector ut. The set of these sub-vectors{vt}Tt=1are concatenated 380
to obtain the bit-vector v, which comprises b¯n bits. 381
Finally, the IrURC-encoded bit vector v is interleaved byπ2 382
in order to obtain the bit vector w, which is modulated onto 383
the uncorrelated non-dispersive Rayleigh fading channel using 384
Gray-mapped QPSK. The overall effective throughput of the 385
proposed scheme is given by η = RIrUEC· RIrURC· log2(M), 386
where we have M= 4 for QPSK. 387
III. IRUEC-IRURC DECODER 388
In this section, we introduce the receiver of the proposed 389
IrUEC-IrURC scheme shown in Fig. 1. In analogy with the 390
IrURC encoder, the IrURC decoder employs T number of 391
component URC decoders{URCt}Tt=1, each having a distinct 392
independent trellis structure. By contrast, the IrUEC employs 393
a unary decoder and a novel IrTrellis decoder relying on a sin-394
gle irregular trellis. In Section III-A, the demodulator and the 395
iterative operation of the IrURC and IrUEC decoders will be 396
discussed, while in Sections III-B and III-C we will detail the 397
internal operation of two components of the IrUEC decoder, 398
namely of the IrTrellis decoder and of the unary decoder, 399
respectively. 400
A. Demodulator and Iterative Decoding 401
As shown in Fig. 1, QPSK demodulation is employed by 402
the receiver in order to obtain the vector ˜w of Logarithmic 403
Likelihood Ratios (LLRs), which pertain to the bits in the vec-404
tor w. This vector is deinterleaved by π2−1 for the sake of 405
obtaining the LLR vector ˜v, which is decomposed into the T 406
sub-vectors {˜vt}Tt=1 that have the same lengths as the corre-407
sponding sub-vectors of{vt}Tt=1. Here, we assume that a small 408
amount of side information is used for reliably conveying the 409
lengths of all vectors in the IrUEC-IrURC transmitter to the 410
receiver. The sub-vectors{˜vt}Tt=1 are then input to the corre-411
sponding component URC decoders{URCt}Tt=1of the IrURC 412
decoder. 413
Following this, iterative exchanges of the vectors of extrin-414
sic LLRs [21] commences between the Soft-Input Soft-Output 415
(SISO) IrUEC and IrURC decoders. In Fig. 1, the notation ˜u 416
and ˜z represent vectors of LLRs pertaining to the bit vectors 417
u and z, which are related to the inner IrURC decoder and the 418
outer IrUEC decoder, respectively. Additionally, a subscript of 419
this notation denotes the dedicated role of the LLRs, with a, 420
e and p indicating a priori, extrinsic and a posteriori LLRs, 421
respectively. 422
At the beginning of iterative decoding, the a priori LLR vec-423
tor ˜ua is initialised with a vector of zeros, having the same 424
length as the corresponding bit vector u. As shown in the IrURC 425
decoder of Fig. 1, the vector ˜ua is decomposed into the T 426
sub-vectors{˜uat}Tt=1, which have the same lengths as the cor-427
responding sub-vectors of{ut}Tt=1. Together with{˜vat}Tt=1, the 428
sub-vectors{˜uat}Tt=1are fed to the corresponding URC decoder 429 URCt, which then outputs the resulting extrinsic LLR vectors 430 {˜ue
t}tT=1 by employing the logarithmic Bahl-Cocke-Jelinek- 431
Raviv (BCJR) algorithm [22]. These vectors are combined for 432 forming the extrinsic LLR vector ˜ue that pertains to the vec- 433 tor u, which is sequentially deinterleaved by the blockπ1−1in 434 order to obtain the a priori LLR vector ˜za that pertains to the 435 bit vector z. Similarly, the IrTrellis decoder is provided with 436 the a priori LLR vector˜zaand generates the vector of extrinsic 437 LLRs˜ze, which are interleaved in the blockπ1to obtain the a 438 priori LLR vector ˜ua that is provided for the next iteration of 439
the IrURC decoder. 440
B. IrTrellis Decoder 441
As discussed in Section II, our IrUEC code employs a novel 442 bit-based irregular trellis, while the IrURC code employs a 443 selection of independent trellises. The novel IrTrellis decoder 444 within the IrUEC decoder applies the BCJR algorithm to the 445 irregular trellis. The synchronization between the novel irreg- 446 ular trellis and the unary codewords is exploited during the 447 BCJR algorithm’s γt calculation of [22, (9)]. This employs 448 the conditional transition probability Pr(Mj = mj|Mj−1= 449
mj−1), where we have 450 P(mj|mj−1) = P(mj, mj−1) rj ˇm=1P( ˇm, mj−1) (7) and P(mj, mj−1) is given in (5). 451
Note that the IrUEC decoder will have an EXIT function 452 [23] that reaches the(1, 1) point of perfect convergence to an 453 infinitesimally low Symbol Error Ratio (SER), provided that all 454 component codebooks in the set{Cs}sS=1have a free distance of 455
at least 2 [24], as characterised in Section IV. Since the combi- 456 nation of the IrURC decoder and demodulator will also have an 457 EXIT curve that reaches the(1, 1) point in the top right corner 458 of the EXIT chart, iterative decoding convergence towards the 459 Maximum Likelihood (ML) performance is facilitated [25]. At 460 this point, the IrTrellis decoder may invoke the BCJR algorithm 461 for generating the vector of a posteriori LLRs˜ypthat pertain to 462 the corresponding bits in the vector y. 463
C. Unary Decoder 464
As described in [10], the unary decoder of Fig. 1 sorts the 465 values in the LLR vector˜ypin order to identify the a number of 466 bits in the vector y that are most likely to have values of zero. 467 A hard decision vectorˆy is then obtained by setting the value of 468 these bits to zero and the value of all other bits to one. Finally, 469 the bit vector ˆy can be unary decoded in order to obtain the 470 symbol vector ˆx of Fig. 1, which is guaranteed to comprise a 471
number of symbols. 472
IV. ALGORITHM FOR THEPARAMETRIZATION OF THE 473
IRUEC-IRURC SCHEME 474
The performance of the IrUEC-IrURC scheme depends on 475 how well it is parametrized. A good parametrization is one that 476
IEEE
Proof
Fig. 3. The legitimate paths through the first three stages in UEC trellis having the codewordsC = {000, 011}.
results in a narrow but still open EXIT chart tunnel, although 477
achieving this requires a high degree of design freedom, when 478
shaping the IrUEC and IrURC EXIT functions. Therefore, we 479
begin in Section IV-A by characterising the free distance prop-480
erty of the UEC codes and selecting a set of UEC component 481
codes having a wide variety of different inverted EXIT function 482
shapes. This maximises the degree of freedom that is afforded, 483
when matching the IrUEC EXIT function to that of the IrURC 484
code. In Section IV-B, we propose a novel extension to the 485
double-sided EXIT chart matching algorithm of [14], which we 486
employ for jointly matching the EXIT functions of the IrUEC 487
and the IrURC codes. However, in contrast to the algorithm of 488
[14], which does not allow a particular coding rate to be targeted 489
for the IrUEC-IrURC scheme, our algorithm designs both the 490
fractionsα and β to achieve a particular target coding rate. In 491
Section V, this will be exploited to facilitate a fair comparison 492
with benchmarkers having particular coding rates. 493
A. Design of UEC Component Codes 494
Since an r -state n-bit UEC code is parametrized by a code-495
book setC comprising r/2 number of codewords each having 496
n bits, there are a total of 2n·r/2 number of candidates forC. 497
It is neither possible nor necessary to employ all these 2n·r/2 498
codebooks as the component codes in our IrUEC code, because 499
some of the codebooks will have identical or similar inverted 500
EXIT function shapes, offering no additional degree of free-501
dom, when performing EXIT chart matching. Therefore, it is 502
desirable to eliminate these candidate codebooks. 503
The generalised UEC trellis structure associated with the 504
codebook C = {c1, c2, . . . , cr/2−1, cr/2} is depicted in [10, 505
Fig. 3(a)]. Note that the upper half and the lower half of the trel-506
lis is symmetrical in terms of the output codewords zjgenerated 507
in response to a given input bit value yj, as shown in (6). More 508
specifically, for the states in the upper half of the trellis, the 509
output codewords zj are selected from the codebook C when 510
yj = 0, while the codewords from its complementary code-511
bookC = {c1, c2, . . . , cr/2−1, cr/2} are selected when yj = 1. 512
For the states in the lower half of the trellis, the output code-513
words zj are selected from the codebookC when yj = 1 and 514
from the complementary codebookC when yj = 0. Intuitively, 515
if any particular subset of the n bits at the same positions within 516
each codeword of C are inverted, this would not change the 517
distance properties of the output bit vector z, hence resulting 518
in an identical inverted EXIT function. For example, inverting 519 the first bit of each codeword in the codebookC0= {00, 01} 520 will give a new codebook C1= {10, 11} having an identical 521 EXIT function. Likewise, inverting both bits of the codewords 522 inC0will giveC2= {11, 10}, which also has an identical EXIT 523 function. Similarly, swapping any pair of the n bits at the same 524 positions between each pair of codewords will not affect the 525 distance properties or the shape of the inverted EXIT function 526 either. For example, swapping the two bits in the codebookC0 527 results in a new codebookC3= {00, 10}, having an identical 528 inverted UEC EXIT function shape. Therefore, each of these 529 four codebooks,C0,C1,C2andC3, as well as their conversions 530 created by bit-inversion and swapping, have identical inverted 531 EXIT functions. Consequently, all but one of these codebooks 532 can be eliminated as candidates for the sake of reducing the 533
complexity of EXIT chart matching. 534
The number of candidate UEC codebooks may be further 535 reduced by characterising their free distance properties. Since 536 no analytic method has been developed for calculating the free 537 distance df of a UEC code, we propose a heuristic method 538
for obtaining an approximate measure of df. The free dis- 539
tance represents the minimum distance between any pair of 540 encoded bit vectors produced by different paths through the 541 trellis. The total number of possible pairings of paths emerg- 542 ing from a particular state in a UEC trellis of length b is given 543 by 2b−1(2b− 1), which grows exponentially. However, consid- 544 ering the symmetry of a regular UEC trellis, it is possible to 545 use a step-by-step directed search for determining the free dis- 546 tance, rather than using a brute force exhaustive search. Note 547 that in the regular UEC trellis as generalised in [10, Fig. 3(a)], 548 a bit vector y= [yj]bj=1identifies a unique path m= [mj]bj=0 549
that emerges from state 1 and terminates at either state 1 550 or 2, hence accordingly identifying a corresponding output 551 bit sequence z= [zk]bk=1¯n . By exploiting this observation, the 552
free distance df can be obtained by computing the Hamming 553
Distance(HD) between each pair of paths and then selecting the 554 pair having the minimum HD, whenever two paths merge at a 555
particular state in the trellis. 556
When the bit sequence length considered satisfies b> 557 r/2, the paths form complete trellis stages, as exemplified 558 in Fig. 3. Therefore, in order to reduce the search complex- 559 ity, we consider all permutations of the b-bit unary-encoded 560 vector y bit-by-bit, considering all paths that emerge from 561 state m0= 1 and terminate at each particular state mb= 562
1, 2, . . . , r, on a step-by-step basis. For a pair of states 563 mj, mj ∈ {1, 2, 3, . . . , r}, we define d
j
mj,mj as the minimum 564 HD between the set of all paths that terminate at state mj 565
and the set that ends at state mj, given the input bit sequence 566 [y1, y2, . . . , yj], where j∈ {0, 1, . . . , b}. Each state mj is 567
labelled as (dmj j,1, d j mj,2, d j mj,3, . . . , d j mj,r), where we have 568 dmj j,mj = d j
mj,mj. For each state m0∈ {1, 2, 3, . . . , r}, the min- 569 imum HDs are initialized to 0s. Therefore, the distance dmj
j,mj 570 can be calculated by 571 dmj j,mj =mj−1min,mj−1 dmj−1 j−1,mj−1+ h(zmj−1,mj, zmj−1,mj) . (8)
IEEE
Proof
Here, zmj−1,mj is the codeword in the setC or in thecomple-572
mentary set C that is generated by the transition from state 573
mj−1 to state mj, while the function h(·, ·) denotes the HD 574
between the two operands. Owing to this, our method con-575
ceived for determining the free distance of a UEC code has 576
a complexity order of O[b· r(r − 1)], where r is the number 577
of states in the trellis and b is the length of the bit vector y 578
considered. Let Yb1 be the bit sequence set associated with
579
the set of all paths Mb1 having a length of b1, while Yb2 is
580
the bit sequence set associated with the path set Mb2 having
581
a length of b2. Therefore, all sequences in Yb1 are prefix of
582
sequences inYb2, when we have b1< b2. For example, when
583
b1= 2 and b2= 3, the bit sequence y2= {111011} is a prefix 584
of the bit sequence y3= {111011111}, where y2is associated 585
with the path vector m2= {1, 3, 2} and y2is associated with the 586
path vector m2= {1, 3, 2, 1}, respectively. Note that accord-587
ing to [26, Lemma 1], the minimum HD df(Yb1) among all
588
bit sequences inYb1 is an upper bound on the minimum HD
589
df(Yb2) of Yb2, when we have b1< b2. Owing to this, the
590
approximate free distance df calculated using our method con-591
verges to the true free distance, as the lengths of the paths 592
considered are extended towards infinity. In our experiments, 593
we considered bit vector lengths of up to b= 10r. In all cases, 594
we found that the free distance has converged before that point, 595
regardless of how the UEC code is parametrised, owing to the 596
common features of all UEC codes described in Section II-B.” 597
For example, Fig. 3 shows all of the legitimate paths 598
through an r= 4-state trellis employing the codebook C = 599
{000, 011} that may be caused by the first three bits in a 600
bit vector y= {yj}bj=1, having a length b> 3. Particularly, 601
the minimum HD d21,3 between states m1= 2 and m1= 3 602
is given by d21,3 = d10,1+ h(111, 000) = 3. Since there are 603
no legitimate paths leading to the states m1= 1 or m1= 604
4, we do not update the associated distances, as shown 605
in Fig. 3. Similarly, we have d12,2= d21,3+ h(111, 011) = 4, 606
and d13,2 = min(d12,2+ h(000, 111), d12,4+ h(000, 100), d22,3+ 607
h(111, 011), d2
3,4+ h(011, 100)) = 4. Once the forward recur-608
sion has considered a sufficient number of trellis stages 609
for min(d1j,1, d1j,2, d2j,2) = min(d1j,1−1, d1j,2−1, d2j,2−1), then the 610
approximate free distance becomes df = min(d1j,1, d1j,2, d2j,2). 611
Our set of candidate component UEC codes was further 612
reduced by considering their free distances. More specifically, 613
in order to achieve a wide variety of EXIT function shapes, 614
we retained only UEC codebooks having the maximal or min-615
imal free distances for each combination of n∈ {2, 3, 4} and 616
r ∈ {2, 4}, where a free distance of 3 is the minimal value that 617
facilitates convergence to the (1, 1) point [24] and avoids an 618
error floor. We drew the EXIT functions for all remaining can-619
didate component UEC codes and selected the five codebooks 620
offering the largest variety of EXIT function shapes, as listed in 621
Table II. Our experiments revealed that only insignificant EXIT 622
function shape variations are obtained, when considering more 623
than r = 4 states. Without loss of generality, our irregular trel-624
lis example of Fig. 2 is constructed by concatenating the five 625
UEC codebooks of Table II. In the following simulations, we 626
will consider irregular trellises that are constructed using these 627
TABLE II
AFTERINVERTING ANDSWAPPING, WESELECT THEIRUEC COMPONENTUEC CODEBOOKS{Cs}5s=1WITHn BITS ANDr STATES BOTH UP TO4. ALL THECODEBOOKS ARE IN THEFORMAT(Cs, df),
WHEREdf IS THEAPPROXIMATEFREEDISTANCE
Fig. 4. Inverted EXIT functions for the S= 5 component UEC codes {UECs}5
s=1of Table II, when extended to r= 10 states codebooks, and when the symbol values obey a zeta probability distribution having the parameter value p1= 0.797.
codebooks. However, the number of states r employed by our 628 five UEC component codes can be optionally and independently 629 increased in the receiver, in order to facilitate nearer-to-capacity 630 operation at the cost of an increased decoding complexity [10]. 631 This is achieved by repeating the last element in the code- 632 book. For example, while the transmitter may use the codebook 633 C = {00, 01}, the receiver may extend this to the r = 10-state 634 codebook C = {00, 01, 01, 01, 01}. Fig. 4 plots the inverted 635 EXIT functions of the component UEC codes {UECs}5s=1, 636 when extended to r = 10 states. Note that, similar to the IrURC 637 EXIT function, the composite IrUEC EXIT function fIrUEC is 638 given as a weighted average of the component EXIT functions 639
{ fUECs}5s=1, where we have 640
fIrUEC = 5
s=1
αs· fUECs. (9)
B. Double-Sided EXIT Chart Matching Algorithm 641 The sixth column of Table III provides the specific Eb/N0 642 values, where the DCMC capacity becomes equal to the 643 throughputη of each scheme considered. These Eb/N0values 644 represent the capacity bound, above which it is theoretically 645 possible to achieve reliable communication. Note that the 646 capacity bound is a function of the overall effective throughput 647 η of the proposed IrUEC scheme, as described in Section II- 648 C. In turn, the overall effective throughput η depends on the 649
IEEE
Proof
TABLE III
CHARACTERISTICS OF THEVARIOUSSCHEMESCONSIDERED, INCLUDINGOUTERCODINGRATERO, INNERCODINGRATERI
ANDEFFECTIVETHROUGHPUTη. EB/N0BOUNDS AREGIVEN FOR THECASE OFGRAY-CODEDQPSK TRANSMISSIONOVER
ANUNCORRELATEDNARROWBANDRAYLEIGHFADINGCHANNEL. COMPLEXITY ISQUANTIFIED BY THEAVERAGENUMBER OFACS OPERATIONSINCURREDPERDECODINGITERATION ANDPERBIT IN THEVECTORz
Fig. 5. Data-flow diagram of the proposed double-sided EXIT chart matching algorithm.
IrUEC coding rate RIrUEC, which depends on the entropy of the 650
zeta distribution HX, as described in Section II-A. In order to 651
facilitate the creation of an open EXIT chart tunnel, it is nec-652
essary, but not sufficient, for the area Aobeneath the inverted 653
outer EXIT function to exceed the area Ai beneath the inner 654
EXIT function [17]. Therefore, the area bound provides the 655
Eb/N0values where we have Ao= Ai, which would theoret-656
ically allow the creation of an open EXIT chart tunnel [27], 657
if the outer and inner EXIT functions were shaped to match 658
each other. Here, Ao and Ai are the areas beneath the outer 659
and inner EXIT functions, respectively. Depending on how well 660
the EXIT functions match each other, a narrow but open EXIT 661
chart tunnel can only be created at a specific Eb/N0 value, 662
which we refer to as the tunnel bound. Based on these obser-663
vations, the Eb/N0difference between the capacity bound and 664
the area bound quantifies the capacity loss that is mitigated by 665
JSCC, while the difference between the area bound and the 666
tunnel bound quantifies the capacity loss that is mitigated by 667
irregular coding [28]. Based on this observation, our double-668
sided EXIT chart matching algorithm may be iteratively applied 669
in order to match a pair of composite outer and inner EXIT 670
functions, which are formed as a combination of S component 671
UEC EXIT functions and T constituent URC EXIT functions, 672
where the latter depend on the Eb/N0 value of the channel. 673
In this way, a narrow but open EXIT chart tunnel between the 674
inverted IrUEC EXIT function and the inner IrURC EXIT func-675
tion may be created at Eb/N0values that approach the capacity 676
and area bounds, hence avoiding capacity loss and facilitating 677
near-capacity operation. 678
As depicted in the data-flow diagram of Fig. 5, the algorithm 679
commences by selecting the fractions α, in order to yield an 680
IrUEC code design having a particular coding rate RIrUECand 681
a composite IrUEC EXIT function that is shaped to match the 682 average of T URC EXIT functions that correspond to a partic- 683 ular Eb/N0value. The technique of [14] may be employed for 684 selecting the fractionsβ, in order to yield a composite IrURC 685 EXIT function that is shaped to match that of the IrUEC code. 686 Following this, the algorithm alternates between the matching 687 of the composite IrUEC EXIT function to the composite IrURC 688 EXIT function and vice versa, as shown in Fig. 5. In order 689 to facilitate near-capacity operation, we use a 0.1 dB Eb/N0 690 decrement per iteration for the component URC EXIT func- 691 tions, when designing the fractionsβ for the IrURC code, until 692 we find the lowest Eb/N0value that achieves a marginally open 693 EXIT tunnel. Note that the double-sided EXIT chart matching 694 algorithm allows the design of an IrUEC code having a spe- 695 cific coding rate RIrUEC. This enables us to design the IrUEC 696 code to have a coding rate of RIrUEC= 0.254, which provides 697 a fair performance comparison with the regular UEC-IrURC 698 scheme of [10] and with other benchmarkers, as detailed in 699 Section V. More specifically, this results in the same overall 700 effective throughput ofη = RIrUEC· RIrURC· log2(M) = 0.508 701
bit/s/Hz, as listed in Table III. 702
For the IrURC encoder, we employ the T = 10-component 703 URC codes{URCt}10t=1of [20], [29]. After running the double- 704 sided EXIT chart matching algorithm of Fig. 5 until the Eb/N0 705 value cannot be reduced any further without closing the EXIT 706 chart tunnel, the composite EXIT functions of the IrUEC and 707 IrURC schemes are obtained, as depicted in Fig. 6(a). Here, the 708 Eb/N0value is 0.3 dB, which is 0.35 dB away from the DCMC 709 capacity bound of−0.05 dB and was found to be the lowest one 710 that creates an open EXIT chart tunnel. More specifically, the 711 fractions of the bit vector z that are generated by the constituent 712 UEC codes{UECs}5s=1of the IrUEC encoder areα = [0 0.7240 713 0.0924 0 0.1836], respectively. Similarly, the fractions of the bit 714 vector u that encoded by the constituent URC codes{URCt}10t=1 715 of the IrURC encoder areβ = [0.1767 0 0.8233 0 0 0 0 0 0 0], 716
respectively. 717
V. BENCHMARKERS ANDSIMULATIONS 718
In this section, we compare the SER performance of the 719 proposed IrUEC-IrURC scheme of Fig. 1 to that of various 720 SSCC and JSCC benchmarkers. As mentioned in Section IV, 721 the proposed IrUEC-IrURC scheme and all benchmarkers are 722 designed to have the same effective overall throughput of 723
IEEE
Proof
Fig. 6. Composite EXIT functions of (a) the IrUEC decoder employing S= 5 component UEC codes {UECs}5s=1, (b) the EG-IrCC decoder employing the S= 13 component recursive systematic CC codes{CCssys}13s=1and (c) the EG-IrCC scheme employing the S= 11 component non-systematic CC codes {CCsns}11s=1, and the IrURC scheme employing the T= 10 component URC codes {URCt}10t=1, when conveying symbols obey a zeta distribution having the parameter p1= 0.797,
and communicating over a QPSK-modulated uncorrelated narrowband Rayleigh fading channel. The EXIT chart tunnel is marginally open when Eb/N0= 0.3,
2.0 and 1.1 dB, respectively.
Fig. 7. Schematic of the EG-IrCC-IrURC benchmarker, in which an EG-IrCC code is serially concatenated with IrURC code and Gray-coded QPSK modulation schemes. Here,π1andπ2represent interleavers, whileπ1−1andπ2−1represent the corresponding deinterleavers.
η = 0.508 bit/s/Hz, for the sake of fair comparison. A pair 724
of benchmarkers are constituted by the UEC-IrURC and EG-725
CC-IrURC schemes of our previous work [10]. Furthermore, 726
a new benchmarker is created by replacing the unary encoder 727
and the IrTrellis encoder in the transmitter of Fig. 1 with an 728
EG encoder and an IrCC encoder, respectively. This results in 729
the SSCC benchmarker of Fig. 7, which we refer to as the EG-730
IrCC-IrURC scheme. Table I shows the first ten codewords of 731
the EG code, which are used for encoding the symbol vector x. 732
As in the IrUEC-IrURC scheme, the bit vector y output by 733
the EG encoder may be modeled as a realization of vector Y= 734
[Yj]bj=1having binary RVs. However, as observed in [10], these 735
RVs do not adopt equiprobable values Pr(Yj = 0) = Pr(Yj = 736
1), hence giving a less than unity value for the correspond-737
ing bit entropy HYj. Similarly, the bit vector z of Fig. 7 may 738
be modeled as a particular realization of a vector Z= [Zk]bk=1¯n 739
comprising b¯n binary RVs. Each binary RV Zk adopts the val-740
ues 0 and 1 with the probabilities Pr(Zk = 0) and Pr(Zk= 1) 741
respectively, corresponding to a bit entropy of HZk. In the case 742
where the IrCC code employs systematic component codes, the 743 bits of y having the entropy HYj < 1 will appear in z, resulting 744 in a bit entropy of HZk < 1. However, a bit entropy of HZk < 1 745 is associated with a capacity loss, as described in [10]. 746 Hence, for the sake of avoiding any capacity loss, it is 747 necessary to use non-systematic recursive component codes, so 748 that the bits in the resultant encoded vector z have equiprob- 749 able values [10]. In order to demonstrate this, we introduce 750 two versions of the EG-IrCC-IrURC benchmarker. Firstly, 751 the N = 13 recursive systematic component CC codes [15] 752 {CCs
sys}13s=1 that were originally proposed for IrCC encoding 753
are adopted in the EG-IrCC-IrURC encoder, as it will be 754 described in Section V-A. Secondly, Section V-B employs the 755 S= 11 non-systematic recursive CC codebooks {CCsns}11s=1 756 proposed in [20], in order to offer an improved version of the 757 EG-IrCC benchmarker. Meanwhile, the 10 component URC 758 codebooks{URCt}10t=1employed by the IrURC encoder in both 759 versions of the benchmarker of Fig. 7 are identical to those in 760
IEEE
Proof
A. Recursive Systematic Component CC Codes 762
The recursive systematic CC codes {CCssys}13s=1 employed 763
in [15] were designed to have coding rates of RCCs
sys ∈
764
{0.1, 0.15, . . . , 0.65, 0.7}. However, since the EG-encoded bits 765
in the vector y are not equiprobable, none of the system-766
atic bits in the bit vector z will be equiprobable either. As a 767
result, the coding rate RCCs
sys =
HY j nCCssys·HCCssysZk
of each system-768
atic CC will be lower than the above-mentioned values. Since 769
each CC code CCssys produces a different number of system-770
atic bits, each will have a different bit entropy HCC s
sys
Zk , and the 771
EXIT function of each CC code will converge to a different 772 point(HCC s sys Zk , H CCssys
Zk ) in the EXIT chart [30]. The composite 773
IrCC EXIT function will converge to a point(HZIrCC k , H
IrCC Zk ), 774
where HZIrCC
k is given by a weighted average of {H CCssys Zk } 13 s=1, 775 according to 776 HZIrCCk = 13 s=1 αs· HCCssys Zk . (10)
Since the vector z is interleaved to generate the bit vector u 777
as the input of the IrURC encoder, the IrURC EXIT function 778
will also converge to (HZIrCC k , H
IrCC
Zk ). However, this presents 779
a particular challenge, when parametrizing the fractionsα and 780
β of the EG-IrCC(sys)-IrURC scheme. More specifically, the 781
fractionsα vary as our double-sided EXIT chart matching algo-782
rithm progresses, causing the entropy HZIrCC
k to vary as well. 783
This in turn causes the IrURC EXIT function to vary, cre-784
ating a cyclical dependency that cannot be readily resolved. 785
More specifically, the fractionsα must be selected to shape the 786
EG-IrCC EXIT function so that it matches the IrURC EXIT 787
function, but the IrURC EXIT function depends on the fractions 788
α selected for the EG-IrCC EXIT function. 789
Owing to this, we design the fractions α and β by assum-790
ing that the bits of y are equiprobable and by plotting the 791
inverted EXIT functions for the S= 13 recursive systematic 792
CC codes accordingly, giving convergence to the(1, 1) point 793
in Fig. 6(b). Then we invoke our double-sided EXIT matching 794
algorithm to design the fractionsα and β for the IrCC(sys) and 795
IrURC codes, which we apply to the EG-IrCC(sys)-IrURC 796
scheme. For the case where the bits of the vector y have 797
the non-equiprobable values that result from EG encoding, 798
the composite EXIT functions are shown in Fig. 6(b). Here, 799
the effective throughput is η = 0.508 bit/s/Hz and the Eb/N0 800
value is 2.0 dB, which is the lowest value for which an open 801
EXIT chart tunnel can be created. This Eb/N0tunnel bound is 802
2.05 dB away from the DCMC capacity bound of−0.05 dB, 803
owing to the above-mentioned capacity loss. Furthermore, the 804
EG-IrCC(sys)-IrURC scheme has an area bound of 1.72 dB, 805
which corresponds to a capacity loss of 1.77 dB, relative 806
to the capacity bound. The designed fractions for the EG-807
IrCC scheme areα = [0.0620 0.2997 0.0497 0.0004 0.1943 0 808
0.0984 0.1285 0 0 0 0.0002 0.1668], while the fractions for 809
the IrURC code are β = [0.6548 0 0.3452 0 0 0 0 0 0 0], 810
respectively. 811
Fig. 8. Inverted EXIT functions for EG-CC code, for the case where the S= 11 component recursive non-systematic CC codes{CCsns}11s=1are employed, and the symbol values obey a zeta probability distribution having the parameter value p1= 0.797.
B. Recursive Non-Systematic Component CC Codes 812 In order to avoid the capacity loss introduced by the recursive 813 systematic CC codes, we advocate the recursive non-systematic 814 CC codebooks{CCsns}11s=1, which are described by the genera- 815 tor and feedback polynomials provided in [10, Table II]. More 816 specifically, of the 12 codes presented in [10, Table II], we 817 use all but the r= 2, n = 2 code, for the sake of avoiding an 818 error floor. These recursive non-systematic CC codes attain the 819 optimal distance properties [31] subject to the constraint of pro- 820 ducing equiprobable bits Pr(Zj = 0) = Pr(Zj = 1), which is 821
necessary for avoiding any capacity loss. The inverted EXIT 822
functions are plotted in Fig. 8. 823
For the sake of a fair comparison, we apply the double- 824 sided EXIT chart matching algorithm of Fig. 5 again to 825 design the EG-IrCC(nonsys)-IrURC scheme having a coding 826 rate of REG-IrCC= 0.254 and an effective throughput of η = 827 0.508 bit/s/Hz. The composite EXIT functions of the EG- 828 IrCC(nonsys) and IrURC schemes are shown in Fig. 6(c). Here, 829 the fractions of the EG-IrCC scheme areα = [0.8101 0 0.0643 830 0 0 0 0 0.1256 0 0 0], while the fractions of the IrURC code are 831 β = [0.2386 0 0.7614 0 0 0 0 0 0 0], respectively. The EXIT 832 chart of Fig. 8 is provided for an Eb/N0value of 1.1 dB, which 833 is the lowest value for which an open EXIT chart tunnel is cre- 834 ated. As shown in Table III, this Eb/N0 tunnel bound is just 835 1.15 dB away from the DCMC capacity bound of−0.05 dB. 836 This improvement relative to the EG-IrCC(sys)-IrURC scheme 837 may be attributed to the non-systematic nature of the EG- 838 IrCC(nonsys)-IrURC scheme, which has reduced the capacity 839 loss to 1.07 dB, as quantified by considering the difference 840 between the Eb/N0 area bound of 1.02 dB and the capacity 841
bound. 842
C. Parallel Component UEC Codes 843
In order to make a comprehensive comparison, we also con- 844 sider a Parallel IrUEC-IrURC scheme. As shown in Fig. 9, 845 this scheme employs a parallel concatenation of S number 846
IEEE
Proof
Fig. 9. Schematic of the Parallel IrUEC-IrURC benchmarker, in which a parallel IrUEC code is serially concatenated with IrURC code and Gray-coded QPSK modulation schemes. Here,π1andπ2represent interleavers, whileπ1−1andπ2−1represent the corresponding deinterleavers.
Fig. 10. SER performance for various arrangements of the proposed IrUEC-IrURC scheme of Fig. 1, the EG-IrCC-IrURC of Fig. 7, the Parallel IrUEC-IrURC scheme of Fig. 9, as well as the UEC-IrURC and the EG-IrURC schemes of [10], when conveying symbols obey a zeta distribution having the parameter p1= 0.797, and communicating over a QPSK-modulated uncorrelated narrowband Rayleigh fading channel having a range of Eb/N0values. A complexity limit
of (a) unlimited, (b) 10,000 and (c) 5,000 ACS operations per decoding iteration is imposed for decoding each of the bits in z.
of separate UEC trellis encoders to encode the bit vector y, 847
in analogy with the structure of the EG-IrCC scheme. More 848
specifically, the component UEC codes of the Parallel IrUEC 849
encoder are selected from the five constituent codes provided in 850
Table II, while the component UEC codes of the Parallel IrUEC 851
decoder are extended to r = 10 states. The irregular fractions 852
employed by the Parallel IrUEC scheme are the same as those 853
used in our proposed IrUEC scheme. However, in order for 854
each component UEC trellis encoder to remain synchronized 855
with the unary codewords in the bit vector y, it is necessary for 856
each component trellis to commence its encoding action from 857
state m0= 1 and end at state mb= 1 or mb= 2. Owing to 858
this, the subvectors of y input to each component UEC must 859
comprise an integer number of complete unary codewords. The 860
irregular coding fractions can only be controlled at the sym-861
bol level in the case of the parallel IrUEC scheme, rather than 862
at the bit level, as in the proposed IrUEC scheme. Therefore, 863
the corresponding EXIT chart of the parallel IrUEC scheme is 864
not guaranteed to have an open tunnel, when the Eb/N0value 865
approaches the tunnel bound of Table III, hence resulting in a 866 degraded SER performance. However, if the frame length a was 867 orders of magnitude higher, the difference between the symbol- 868 based and bit-based segmentations of the bit vector y would 869 become insignificantly small. As a result, a similar SER per- 870 formance may be expected for the parallel IrUEC scheme in 871 this case. In the following section, we will compare the perfor- 872 mances of the Parallel IrUEC and the proposed IrUEC schemes, 873 using different values for the frame length a. 874
D. SER Results 875
The SER performance of the IrUEC-IrURC, the EG- 876 IrCC(sys)-IrURC and the EG-IrCC(nonsys)-IrURC, UEC- 877 IrURC and EG-CC-IrURC schemes is characterised in Fig. 10. 878 In each case, the source symbol sequence x comprises a= 104 879 symbols, the values of which obey a zeta distribution hav- 880 ing a parameter value of p1= 0.797. As shown above, the 881 parametrizations of the irregular codes in each scheme are 882
